There is a popular [citation needed] children's song called Inchworm[0] that has a chorus that lists powers of two.

    Two and two are four
    Four and four are eight
    Eight and eight are sixteen
    Sixteen and sixteen are thirty-two 

Everyone familiar with this song would know that 16 and 16 are 32.

[0] https://en.wikipedia.org/wiki/Inchworm_%28song%29

Complete lyrics (is DMCA coming at me?):

  Inchworm, inchworm, measuring the marigolds
  You and your arithmetic, you'll probably go far.
  Inchworm, inchworm, measuring the marigolds
  Seems to me you'd stop and see how beautiful they are.
  
  (Kids Singing: 2 & 2 are 4, 4 & 4 are 8, 8 & 8 are 16, 16 & 16 are 32)

Nothing to do with imperial units at all.

Even if she didn't know the song, she might have learned 16+16 from those.

There's a similar song in Spanish, although it takes some detours. The maths in the song (which rhyme in Spanish) are the following:

2+2=4

4+2=6

6+2=8

...+8=16

...+8=24

...+8=32

Whole lyrics: http://www.teocio.es/portal/entretenimiento/canciones-danzas...

This is how i often work with things. I break them down into things that i know how to work with. I search my knowledge for elements that most closely represents the problems and then combine them.

For example, rather than remembering 12 * 6 = 72, i recall that 10 * 6 = 60 and 2 * 6 = 12, hence 60 + 12 = 72. I don't need to remember as many things and for me i can do that much faster than trying to recall it.

My memory has never been very good, so perhaps this is why i take such an approach, as i find it overall easier.

That's how I felt about math for a long time. The so-called good laziness. You don't bother doing the hard way, you search for tricks. Symmetries, difference from simple cases as the girl did.

Instead of memorizing algorithms, kids should be shown this. It's free gamification and, after some years, I believe it's a core idea of what is, and why math can be pleasing.

Parents can do it and it can really help.

I actually was taught this and a number of other "tricks" in primary school. Estimation was a defined thing, and the idea of getting quick math results in your head was a real goal. Reasons given: job interview, pricing goods and services to understand value, fuel consumption, making things (that deck, how many boards, cost, etc...), navigation.

They took us through a lot of those cases. The first time I applied this was powers of 2 for computing. Learn the first 16 bits worth, and that helps with all sorts of things in computing, same as powers of ten and common easy to compute things do in most other areas of life.

So, what I did was make it a car game. Figure out how tall things might be, or prices, whatever comes up. Practice doing it helps to actualize the skill and once it's done, they will apply in in ways they find useful.

* is a special character on HN you want to have spaces between 12 * 6 or it can look like: 126 106.

Note: it works in pairs so the final 2*6 shows up.

Ah, thanks :D

Incidentally, this is now explicitly taught in elementary school math classes. They call it 'number bonds' and it often drives parents crazy because "OMG IT ISN'T HOW I LEARNED IT!", but it is clearly a better strategy than 'just memorize this multiplication table'

Remembering the multiplication table up to 12 seems to be an American thing. We only did up to 9 and from then on learned how to multiply many-digit numbers going digit by digit.

It might "just" (not to be little it) be shortcut she has learned, I also used tons of my own shortcuts for answers when I was smaller (read: used to not use calculator). I could do smaller calculations like that fast and when my parents asked how I came up with them and when I answered they didn't believe me.

I think parent wondered how it's possible that she remembered 16+16=32.

Children raised on positional decimal arithmetic are "supposed" to figure that 10+10=20 or 20+20=40 and then add/subtract 5+5.

Of course it's easier to associate 15 with 16 than with 10 or 20, but the fact that she immediately knew 2·16 and was able to proceed further says something about either her experience with binary arithmetic or some tendency to spontaneously count in binary.

I don't know, I always remembered things like 6+7 = 13 and if numbers I was adding were close to 6 and 7 I knew instantly how to proceed. Of course there's other tricks like, subtracting the 1 out of 6 to get 5 and then adding it to 7 making 8 then removing the 5 to get 10 with the other 5 and you are left with 10 and 3 which is easy, I guess.

At least for me it was just silly shortcuts that stuck with me like 6+7=13 or 7*7=49. Maybe why I'm little fixated to 7 is that 5 and below is easy, but with 6 and up things get little more "math-y", so with some quick shortcuts like these it's easy to adjust to 7. Or most likely I'm just full of shit, that is usually the case.

I think that's what you get with unsupervised learning.

Tricks only get you so far, you can't do 6+7 on fingers or 7·7 as (4+3)(4+3) (I had a huge problem with this one, too many numbers to track). So you fail. And when you fail, you memorize the problematic cases and go on with tricks based on those.

And yes, it's completely incomprehensible for people who learned the pencil-and-paper algorithms and think that those are the end of the world.

The real trick is nothing more than the distributive property.

  (2)(16) = (2)(15 + 1)

"Mathematical properties" are just hacks that become popular because they generalize. [0]

[0] http://slatestarcodex.com/2014/03/03/do-life-hacks-ever-reac...

In the decimal system it is just as easy to associate 5+5=10.

5xN is easier to remember than 16xN on account of 5 being the smaller number.

2^n is logarithmic whereas 5*n is linear. Arguably, logarithms are not too complicated, even if linear seems to be a degree easier, seeing that the decimal system is also logarithmic as that's a denser representation.

edit: how to enter an aterisk as the multiplication operator sign

> 5xN is easier to remember than 16xN on account of 5 being the smaller number.

By that logic, shouldn't it be easier to remember 7 * n than 10 * n, because 7 is the smaller number?

> 2^n is logarithmic

Also, as Jtsummers (https://news.ycombinator.com/item?id=10973381) points out, the function `n \mapsto 2^n` is exponential: its growth is significantly faster, not significantly slower, than exponential.

I didn't talk about growth, i was specifically thinking about a comment that called our number system logarithmic and it stands to reason, if you wan't to find x in 10^x=y, unless you do calculus in your head at the age of five, you may as well look at a grap of the exponential progression. now that's two incoherent arguments, but the latter should point out that it doesn't really matter which way, and the first alludes that there is something more basic to the logarithm, something easier to capture. If I look at the grid pattern in logarithmic plots, it's not a runaway progression but a nice repetition where the zeros in the end of the numbers at each 10^n interval behave like the unary number system, which is the basic positional system and arguably easier even then binary.

Please read 'linear' in place of the last word 'exponential' in my reply. Oops.

Re edit: Put a space between the * and the characters.

  5 * N

versus

  5*N

Pairs of asterisks adjacent to non-space characters become markers indicating italicized text.

  5*N is easier to remember than 16*N

5N is easier to remember than 16N

Versus

  5 * N is easier to remember than 16 * N

5 * N is easier to remember than 16 * N

And I think the word you want is exponential, not logarithmic. Related to each other, since exponential expressions become linear on a logarithmic scale, but exponential describes the growth of 2^n better.

It may be that the girl happened to know 16+16=32 because she had answered the question before, and it stuck in her memory.

As much as I regret that humans don't have 8 fingers at each hand and count in base 16, I think it's more likely that she simply is/knows a computer nerd.

Maybe the child has simply played a doubling game before, along these lines, and memorized it:

What's 1 and 1? 2.

What's 2 and 2? 4.

What's 4 and 4? 8.

What s 8 and 8? 16.

What's 16 and 16? 32.

That's just five facts.

The game could occur socially between kids. It is natural to ask a question, then take the answer and "up the ante" by re-formulating the answer into a harder question, back into the other child's face!

Do you know 1 plus 1? Ha, two plus two? Oh yeah, how about four plus four, then?

The progression grows quickly, of course, and soon the interrogated subject breaks.

I get beaten at 65536.

Ah, if only "one twenty eight kay" were an accepted answer ... :)

It's also doubling. Like ashark, I tended to play with numbers like that as a kid. Plus, not sure how old I was but I think I was 7, perhaps 8, I was introduced to the question:

  If I agreed to pay you a dollar today, and double your pay
  each day. How much would you get paid after a week? After
  30 days?

By a teacher (presented differently, but that's the question). The power of doubling just stuck with me (how quickly it grew), long before I actually knew the concepts of linear versus exponential growth. This resulted in certain arithmetic facts sticking with me better than other, perhaps more logically obvious facts. Random things like this will stick with kids when they can start to see patterns or features within the structure of it.

I used to run through this (and on up to higher powers of 2) in my head as a kid in idle moments, well before I knew it had any relevance to... anything, really.

If you learn Egyptian method of counting you can get to 12 on 1 hand and with a single small tweak to it. We could count to 16 on each hand. If we taught kids how to count differently

I remember knowing 16+16=32 because of doing doubles starting with 1. 1+1=2, 2+2=4, 4+4=8, 8+8=16... if I had to guess, I'd say this is how that little girl knew.

A lot of cooking, at least in the US, is base 2.

Yeah, I think imperial units are good explanation as well, if this is what you meant.

Maybe finds it easier to remember one large addition and to correct it with smaller additions.

More likely she's played 2048 a couple times. Obligatory xkcd:

https://xkcd.com/1344/